Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{q^2 - 5q}{q^2 - 8q + 15}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 5q}{q^2 - 8q + 15} = \dfrac{(q)(q - 5)}{(q - 3)(q - 5)} $ Notice that the term $(q - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 5)$ gives: $x = \dfrac{q}{q - 3}$ Since we divided by $(q - 5)$, $q \neq 5$. $x = \dfrac{q}{q - 3}; \space q \neq 5$